Answer
$ S_{10}=1705 $
$S'_{10}=-575$
$ S_{10}+S'_{10}=1705-575=2280$
Work Step by Step
We are given that the sequence $ a_n $ is a geometric sequence and the sequence $ b_n $ is an arithmetic sequence.
Here, we have $ r=-2$ and $ d=-15$
The general formula to find the sum of first n term of a Geometric sequence is given as: $ S_{n}=\dfrac{a_1(1-r^n)}{(1-r)}$
Thus, $ S_{10}=\dfrac{(-5) \times (1-(-2)^{10})}{[1-(-2)]} \\=\dfrac{(-5) \times (1-2^{10})}{[1+2]}=1705 $
The general formula to find the sum of the first n term of an Arithmetic sequence is given as: $ S'_{10}=\dfrac{n(b_1+b_n)}{2}$
and $ S'_{10}=\dfrac{10(b_1+b_1+(n-1)d)}{2} \\=\dfrac{10(10+10+(10-1) (-15))}{2}=-575$
Now, $ S_{10}+S'_{10}=1705-575=2280$
